The most common greeting used in Korea is "안녕하세요?" just like "Hello." in English.
It is used to greet someone you know or someone you meet for the first time.
안녕하세요.(an-nyong-ha-seh-yoh) is a polite way to say hello. It is used to greet someone who is older than you or someone you have to treat politely.
안녕(an-nyong) is a casual form of hello. You use it to someone who is the same age as you or younger than you.
안녕하십니까?(an-nyong-ha-sip-nee-ka) is also a polite form of hello and mostly used in official forms or business. You won't hear this greeting in a casual conversation.
There are two ways to say goodbye in Korean. One is to one who "leaves" the place and the other is to one who "stays".
Knowing the verbs "to go"(가다) and "to stay"(있다., 계시다) will help you to memorize this easily.
안녕히 가세요.(annyong-hee-ka-se-yo) is a polite term of goodbye to one who leaves the place.
잘 가요.(jahl-ka-yo) is a semi-polite way of saying goodbye to one leaves.
잘 가(jahl-ka) is a casual way to say goodbye to one who leaves.
안녕히 계세요.(annyong-hee-ke-se-yo) is a polite term of goodbye to one who stays.
잘 있어요.(jalit-sseo-yo) is a semi-polite way of saying goodbye to one stays.
잘 있어(jalit-sseo) is a casual way to say goodbye to one who stays.
Another way to say goodbye is see you tomorrow or see you next time.
내일 뵈요.(nail-beh-yo) is a semi-casual form of see you tomorrow.
내일 봐 or 내일 만나(nae-il-bwa or nae-il-mahnna) is a casual term of saying see you tomorrow.
또 뵙겠습니다.(ttoh-bep-ket-seumnee-da) is a polite term of I’ll be seeing you again.
다음에 봐 or 또 보자(da-eumeh-bwa or tto-bo-ja) is a casual term of see you again or let's meet again.
A verb "죄송하다" is used to express one's apology to people who's older or higher position than the speaker. "미안하다" is for a person of the same age or someone who's younger than the speaker. "잘못했어요" is more commonly used when you've done something wrong or made a mistake.
죄송합니다 or 잘못했습니다(je-song-hamnee-da or jal-mottet-seumnee-da) is a polite term of "I'm sorry".
죄송해요(je-song-hae-yo) is a semi-casual way to say "I'm sorry".
미안해요 or 미안합니다(mee-an-hae-yo or mee-an-hamnee-da) is a semi-casual way to say "I'm sorry".
미안해 or 잘못했어(mee-an-hae or jal-motte-sseo) is a casual way to say good night.